Physics 2102 QuickTime and a decompressor are needed to see this picture Jonathan Dowling Lecture 21 THU 01 APR 2010 Ch 31 4 7 Electrical Oscillations LC Circuits Alternating Current QuickTime and a decompressor are needed to see this picture QuickTime and a decompressor are needed to see this picture LC Circuit At t 0 1 3 Of Energy Utotal is on Capacitor C and Two Thirds On Inductor L Find Everything Phase 0 1 2 U B t L q0 sin t 0 2 1 2 U E t q0 cos t 0 2C 2 2 1 L q sin 0 0 U total 3 U BB 0 2 00 total 2 2 1 2 U total 3 2U U EE 0 total q00 cos 0 2C tan 0 1 2 0 arctan 1 2 35 3 1 2 U B 0 L q0 sin 0 U total 3 2 1 2 U E 0 q0 cos 0 2U total 3 2C 2 LC q0 sin 0 1 1 LC 2 2 q0 VC q0 cos 0 q q0 cos t 0 i t q0 sin t 0 i t 2q0 cos t 0 q VL t 0 cos t 0 C q0 VC t cos t 0 C Damped LCR Oscillator Ideal LC circuit without resistance oscillations go C on forever LC 1 2 L R Real circuit has resistance dissipates energy oscillations die out or are damped Math is complicated Important points Frequency of oscillator shifts away from 1 0 LC 1 2 0 8 QLCR 2L R For small damping peak ENERGY decays with 0 6 UE Peak CHARGE decays with time constant U max Q2 RtL e 2C 0 4 time constant 0 2 ULCR L R 0 0 0 4 8 12 time s 16 20 DampedOscillationsinan RCL Circuit If we add a resistor in an RL circuit see figure we must modify the energy equation because now energy is dU i 2 R dt q 2 Li 2 dU q dq di U U E U B Li i 2R 2C 2 dt C dt dt being dissipated on the resistor dq di d 2 q d 2q dq 1 i 2 L 2 R q 0 This is the same equation as that dt dt dt dt dt C d 2x dx of the damped harmonics oscillator m 2 b kx 0 which has the solution dt dt k b2 x t xm e cos t The angular frequency 2 m 4m For the damped RCL circuit the solution is bt 2 m q t Qe Rt 2 L cos t The angular frequency 1 R2 2 LC 4 L 31 6 q t Q Qe q t Qe Rt 2 L cos t Rt 2 L q t 1 R2 2 LC 4 L Q Qe Rt 2 L The equations above describe a harmonic oscillator with an exponentially decaying amplitude Qe Rt 2 L The angular frequency of the damped oscillator 1 R2 1 2 is always smaller than the angular frequency of the LC 4 L LC R2 1 undamped oscillator If the term we can use the approximation 2 4L LC RC RC RL L R RCL 2L R Summary Capacitor and inductor combination produces an electrical oscillator natural frequency of oscillator is 1 LC Total energy in circuit is conserved switches between capacitor electric field and inductor magnetic field If a resistor is included in the circuit the total energy decays is dissipated by R Alternating Current To keep oscillations going we need to drive the circuit with an external emf that produces a current that goes back and forth Notice that there are two frequencies involved one at which the circuit would oscillate naturally The other is the frequency at which we drive the oscillation However the natural oscillation usually dies off quickly exponentially with time Therefore in the long run circuits actually oscillate with the frequency at which they are driven All this is true for the gentleman trying to make the lady swing back and forth in the picture too Alternating Current We have studied that a loop of wire spinning in a constant magnetic field will have an induced emf that oscillates with time E Em sin d t That is it is an AC generator AC s are very easy to generate they are also easy to amplify and makes them easy to send in distribution decrease in voltage This in turn grids like the ones that power our homes Because the interplay of AC and oscillating circuits can be quite complex we will start by steps studying how currents and voltages respond in various simple circuits to AC s AC Driven Circuits 1 A Resistor emf vR 0 v R emf E m sin d t vR Em iR sin d t R R QuickTime and a decompressor are needed to see this picture Charles Steinmetz Resistors behave in AC very much as in DC current and voltage are proportional as functions of time in the case of AC that is they are in phase For time dependent periodic situations it is useful to represent magnitudes using Steinmetz phasors These are vectors that rotate at a frequency d their magnitude is equal to the amplitude of the quantity in question and their projection on the vertical axis represents the instantaneous value of the quantity AC Driven Circuits 2 Capacitors vC emf E m sin d t qC C emf CE m sin d t dqC iC d CE m cos d t dt iC d CE m sin d t 90 0 Em iC sin d t 90 0 X 1 where X reactance d C Em im X V looks like i R Capacitors oppose a resistance to AC reactance of frequency dependent magnitude 1 d C this idea is true only for maximum amplitudes the instantaneous story is more complex AC Driven Circuits v L emf E m sin d t 3 Inductors d iL v L dt v L L iL dt L Em Em iL cos d t sin d t 90 0 L d L d Em iL sin d t 90 0 X Em im where X L d X Inductors oppose a resistance to AC reactance of frequency dependent magnitude d L this idea is true only for maximum amplitudes the instantaneous story is more complex Transmission lines Erms 735 kV I rms 500 A Step down transformer110 V Step up transformer T1 Home R 220 T2 l 1000 km Power Station Energy Transmission Requirements Ptrans iV big Pheat i2R small Solution Big V l R is fixed 220 in our example A 2 I rms R This parameter is also fixed The resistance of the power line R Heating of power lines Pheat 55 MW in our example Power transmitted Ptrans ErmsI rms 368 MW in our example In our example Pheat is almost 15 of Ptrans and is acceptable To keep Pheat we must keep I rms as low as possible The only way to accomplish this is by increasing Erms In our example Erms 735 kV To do that we need …
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